Dynamics of a Massive Black Hole at the Center of a Dense Stellar System

We develop a simple physical model to describe the dynamics of a massivepoint-like object, such as a black hole, near the center of a dense stellarsystem. It is shown that the total force on this body can be separated into twoindependent parts, one of which is the slowly varying influence of theaggregate stellar system, and the other being the rapidly fluctuatingstochastic force due to discrete encounters with individual stars. For theparticular example of a stellar system distributed according to a Plummermodel, it is shown that the motion of the black hole is then similar to that ofa Brownian particle in a harmonic potential, and we analyze its dynamics usingan approach akin to Langevin's solution of the Brownian motion problem. Theequations are solved to obtain the average values, time-autocorrelationfunctions, and probability distributions of the black hole's position andvelocity. By comparing these results to N-body simulations, we demonstrate thatthis model provides a very good statistical description of the actual blackhole dynamics. As an application of our model, we use our results to derive alower limit on the mass of the black hole Sgr A* in the Galactic center.Comment: 27 pages, 4 figures. ApJ, in press (2002). New section with application to SgrA